The link with billiards arises because a billiard shot in an elliptical billiard corresponds to a point on an elliptic curve which is a member of an elliptic scheme denoted \({\mathcal {L}}\) ; periodicity corresponds to this point being torsion. This correspondence too goes back to long ago, essentially to Jacobi's proof of a famous theorem of Poncelet; see also the book [1]} by V. Dragovic and M. Radnovic for a detailed study of this, also in a more general context of hyperelliptic Jacobians, and see the papers [2]} by W. Barth and J. Michel, and [3]} by B. Jakob. We especially point out the book by J. Duistermaat [4]}, which (see Ch. 11) treats in detail some issues of elliptic surfaces with reference to billiards. But to our knowledge no systematic application of this correspondence, similar to what we propose, has been yet developed or pointed out so far in the literature. Also, we shall develop several details and equations for the basic setting which seem not to appear in full in the existing literature, and this study will involve the investigation of relevant (elliptic) algebraic surfaces and corresponding results in their realm which appear to be new. (The interesting book [4]} contains some results and formulae which we develop independently in the Appendix, however the viewpoint of [4]} is somewhat different and no arithmetical information is introduced.)
